The difference between a stepbrother and a half-brother is whether he is related only through marriage or whether he is a blood relative. Half-brothers share one biological parent, while stepbrothers do not.
A stepbrother is your stepfather's or stepmother's son through a previous relationship. For instance, if your father married a woman who already had a son or if your mother married a man who already had a son, then that boy would be your stepbrother.
However, in order to have a half-brother, you must have either the same mother or father as he does. This means that if your father or mother remarried and had a son, than that boy would be your half-brother.